Re: subrings of ZxZ
 From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
 Date: Sun, 12 Oct 2008 02:53:45 +0000 (UTC)
In article <518cbdf5ae9c4590a023eec81e4a9ded@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Luke Wu <LookSkywalker@xxxxxxxxx> wrote:
What are all the subrings of ZxZ if we require (0,0) and (1,1) to be
in the subring?
the whole ring is a subring
subset of all pairs of form (a,a) is a subring
I believe that's all. Am I right?
No. For example, what about the collection of all pairs (a,b) in which
ab is a multiple of 5?
This contains (0,0) and (1,1), and (a,a) for all a. If b=a+5k and
d=c+5m, then (a,b) + (c,d) = (a,a+5k) + (c,c+5m) = (a+c,(a+c)+5(k+m)),
so the set is closed under sums. It is also closed under differences;
what about products?

======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
 Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidinatmemberamsorg
.
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 subrings of ZxZ
 From: Luke Wu
 subrings of ZxZ
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